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Complexity
Volume 2019, Article ID 7896946, 18 pages
https://doi.org/10.1155/2019/7896946
Research Article

Threshold Dynamics in a Periodic Three-Patch Rift Valley Fever Virus Transmission Model

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

Correspondence should be addressed to Zhidong Teng; moc.anis@gnet_gnodihz

Received 29 April 2018; Revised 17 August 2018; Accepted 29 August 2018; Published 9 January 2019

Academic Editor: Peter Giesl

Copyright © 2019 Buyu Wen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates a three-patch Rift Valley fever virus transmission model with periodic coefficients. The basic reproduction number is calculated for the model by using the next infection operator method. The threshold conditions on the extinction and permanence in the mean of the disease with probability one. The Rift Valley fever disease will be permanent in the -th patch if and dies out in the -th patch if . The numerical simulations are given to confirm the theoretical results.

1. Introduction

Rift Valley fever (RVF) is an acute viral zoonosis caused by the Rift Valley fever virus (RVF virus). This disease is primarily transmitted among domestic animals, such as sheep, cattle, buffalo, goat, and camels, with the main route of infection being mosquito bites (see [14]). RVF virus infection in livestock animals can produce high rate of abortion in pregnant animals and significant morbidity and mortality rate in young ruminants. During the disease epidemic among animals, humans may be infected with Rift Valley fever virus through bites from infected mosquitoes or contacts with the blood, body fluids, and organs of infected animals (see [5, 6]).

In the 1930s, RVF virus was discovered for the first time near Naivasha Lake in the Rift Valley of Kenya. In 1977, a human RVF virus epidemic is reported in Egypt, which is the largest epidemic. Subsequently, RVF virus outbreaks have been reported in Saharan and North Africa. After that, this disease occurred in Saudi Arabia and Yemen outside the Africa in the 2000s. These disease outbreaks result in high mortality and abortion in young ruminants and case significant economic losses in Africa and Middle East. With the enlargement of scope of RVF virus infection, there is a growing concern that this disease will outbreak further in other parts of Asia and Europe. In recent years, numerous mathematical models have been developed to investigate RVF virus outbreaks (see [1, 3, 615]).

Research results show that, in Northeastern Africa, human activities, including those associated with the Eid al-Adha feast, along with a combination of climatic factors such as rainfall level and hydrological variations, contribute to the transmission and dispersal of the disease pathogen. Moreover, sporadic outbreaks may occur when the two events occur together: (1) abundant livestock is recruited into areas at risk from RVF due to the demand for the religious festival and (2) abundant numbers of mosquitoes emerge. In [8, 14, 15], the authors consider the religious festival and discuss three-patch model for RVF virus transmission.

On the one side, a religious festival, the Eid al-Adha feast, at which time large numbers of livestock are driven towards the site of the feast. In Africa and the Middle East, the invasion path follows the same route that people use to travel to the Nile Delta, where the important Islamic festival Greater Bairam is held. Geographically, the disease seems to be along a path from southwest to northeast of Africa. This is consistent with the way to Mecca, Saudi Arabia’s capital (see [15]).

On the other side, the number of mosquitoes is affected by temperature and humidity, the greater the humidity, the more the number of mosquitoes, the humidity at about thirty degrees is the most suitable for mosquitoes to grow and propagate, and lower than ten degrees will enter hibernation. Due to rainfall and temperature, in fact, the mosquito population densities follow seasonal fluctuations, achieving maximum numbers during the rainy season and reaching lowest level during the winter season (see [4, 7, 8]). The abundant numbers of mosquitoes corresponds to an increased number of infected ruminants. Consequently, RVF virus infections fluctuate over time and highly link to seasonal variations.

The autonomous models have been discussed in [3, 1115]. At the same time, we can see that Rift Valley fever will change periodically with the religious festivals and mosquitoes’ periodicity. Recently, in [8], Xiao et al. proposed a three-patch periodic model for RVF virus transmission with the periodic recruitment rate of livestock in first patch, the periodic carrying capacities for mosquitoes in three patches, the periodic natural death rate for livestock in the last patch, and the periodic migration rate. In order to seek the seasonal and festival-driven impacts for RVF outbreaks, the authors in [8] investigated the dynamical behaviors of the model just by using the numerical simulation method.

The main purpose of this paper is to discuss the threshold dynamics of a periodic three-patch Rift Valley fever virus transmission model, and thereout, we will make up the deficiency for the research given in [8]. By applying the methods in given [1621], we calculate basic reproduction number in -patch of model By using the theory of persistence for dynamical systems, we will establish that serves as a threshold value for Rift Valley fever virus dynamics in periodic case.

The organization of this paper is as follows. In the next section, we develop a periodic three-patch model to describe the transmission of Rift Valley fever virus and present some useful lemmas. In Section 3, we will propose the main results, namely, we will obtain the extinction or permanence of all positive solutions of model . In Section 4, we will give the numerical simulations to verify our theoretical results.

2. Model

We investigated the festival-driven and seasonal impacts on the patterns of RVF outbreaks among livestock in Africa and Middle East.

We assume that a disease invades and subdivides the target livestock into four classes: the susceptible class , the exposed class , the infectious class , and the recovered class ; the female mosquitoes are divided into three compartments: uninfected compartment , exposed compartment , and infected compartment in each patch.

We assume that is the number of livestock imported daily at time in 1-patch, is the transmission rate of disease from mosquitoes to livestock at time in -patch, is the transmission rate of disease from livestock to mosquitoes at time in -patch, is the time-dependent natural death rate for livestock in -patch (average death rate for different livestock, i.e., cattle and sheep), is the time-dependent natural death rate for female mosquitoes in -patch, is the rate at which immunity is lost after recovery for livestock at time in -patch, is the rate of becoming infectious for livestock at time in -patch, is the rate of becoming infectious for mosquitoes at time in -patch, is the recovery rate of leaving the infectious class for livestock at time in -patch, is the time-dependent birth rate for mosquitoes in -patch, is the time-dependent carrying capacity of mosquitoes in -patch, is the disease-induced death rate for livestock at time in -patch, is moment speed of animals from -patch to -patch, and is the length of journey for animals within patch . Motivated by the above assumption, we propose the following periodic three-patch RVF virus transmission model. where .

For any continuous and bounded function defined for , we denote and .

Based on the biological background of model , the initial condition for model (for any ) has in the following form:

For model , we give assumptions as follows:

functions and are continuous and -periodic defined on . Furthermore, functions , , , , , , , and are positive, and functions , , , and are nonnegative.

and for all .

Consider a -periodic linear equation as follows: where and are -periodic continuous functions defined for . We have the following lemma.

Lemma 1 (see [17]). Assume that and for all . Then, equation (3) has a unique positive -periodic solution which is globally attractive.

Furthermore, we consider a -periodic logistic equation as follows: where and are -periodic continuous functions defined for . We have the following lemma.

Lemma 2 (see [18]). Assume that and for all . Then, equation (4) has a unique positive -periodic solution which is globally attractive.

When and , we obtain the disease-free subsystem of model :

As consequence of Lemma 1 and Lemma 2, we have the result on the existence of disease-free periodic solution of model .

Lemma 3. System (5) has a unique positive -periodic solution which is globally attractive.

From Lemma 3, model has a unique disease-free periodic solution .

Let be a continuous, cooperative, irreducible, and -periodic matrix function; we consider a linear system:

Let be the fundamental solution matrix of system (6) satisfying initial condition , where is an unit matrix. Let be the spectral radius of matrix . Since is continuous, cooperative, and irreducible, we obtain that is nonnegative for all . By Perron-Frobenius theorem, is the principal eigenvalue of in the sense that it is simple and admits a positive eigenvector . We present the following lemma.

Lemma 4 (see [19]). Let . Then, there exists a positive -periodic function such that is a solution of system (6).

Now, we calculate the basic reproduction number of model by applying the next-generation matrix approach which is given in [20]. Firstly, we easily validate that model satisfies the conditions given in [20]. Let

Let with be the evolution operator of linear -periodic system:

Then for any and and , where is the unit matrix. Let be the ordered Banach space of all periodic continuous functions . Assume that is the initial distribution of infectious individuals. Then, is the rate at which new infections are produced by infected individuals who were introduced into the population at time . When , then represents the distribution of those infected individuals who were newly infected individuals at time and remain in the infected compartments at time . Hence, the cumulative distribution of new infections at time produced by all those infected individuals introduced before is given by a linear operator as follows:

Applying the results obtained in [20], the basic reproduction number for model is defined as spectral radius of operator , that is, .

Furthermore, using Theorem 2.2 given in [20], we also can obtain the following results on basic reproduction number .

Lemma 5 (see [20]). The following conclusions hold. (a) if and only if (b) if and only if (c) if and only if (d)Disease-free periodic solution for model is locally asymptotically stable if and unstable if

Remark 1. In particular, when periodic systems (1) degenerate into autonomous systems, by the next-generation matrix method (see [22]), we can obtain where and . Then,

3. Main Results

For model , we investigate the persistence and extinction of positive solutions on the three patches.

3.1. The First Patch

In this section, we discuss model . Firstly, on the nonnegativity, positivity, and boundedness of solutions for model , we have the following result.

Theorem 1. Let be the solution of model with initial condition (2). Then, is nonnegative for all and ultimately bounded, and when and , then is also positive for all .

Proof 1. In fact, by the continuous dependence of solutions with respect to initial values, we only need to prove that when and , then is positive for all . From the fifth equation of model , we have for any . Since , we get

Define . Obviously, . We only need to prove for all . Suppose that there exists a such that and for all . Then, we only need to discuss the following six cases: (1) , (2) , (3) , (4) , (5) , and (6) .

Now, we only give the proof for case (2). The remaining cases can be proved in a similar manner. Let . Since for all , then , , and hence,

Integrating from to , we get which leads to a contradiction.

Next, we show the boundedness of nonnegative solutions of model . Let , then we get

Hence, we have which implies that and are ultimately bounded. Since for any , using Lemma 3; we further have Hence, for any , there is a such that and for all , where . From the sixth equation of model , we have for all , where . Hence, we can obtain From this, for any , there is a such that for all . Hence, from the last equation of model , we further have for all , where . This shows that . From above discussions, we finally obtain that all solutions of model with initial condition (2) are ultimately bounded. This completes the proof.

Now, we show that serves as a threshold parameter. We prove that when , then disease-free periodic solution is globally asymptotically stable, that is to say, the RVF virus is extinct in the first patch, and when , then model is permanent, that is to say, the RVF virus is existent in the population in the first patch.

Theorem 2. If , then disease-free periodic solution of model is globally asymptotically stable, and if , then is unstable.

Proof 2. From conclusion of Lemma 5, it follows that if , then is unstable, and if , then is locally asymptotically stable.

We now prove that if , then is globally attractive. Let , from conclusion of Lemma 5, then . We can choose constant small enough such that , where

From model , we have

From Lemma 3, for above given , there exists a such that and for any . Hence, from model , we obtain that for all

Consider the following auxiliary system:

It follows from Lemma 4 that there is a positive -periodic function such that is a solution of system (23), where .

Denote We can choose constant such that According to the comparison theorem of the vector form (see [23]), we obtain for any . By , we know Then, we conclude that , that is to say, . By the equations of , , and of model , we also obtain . Hence, disease-free periodic solution is globally attractive. This shows that disease-free periodic solution is globally asymptotically stable. This completes the proof.

Theorem 3. If , then the disease in model is permanent.

Proof 3. From conclusion of Lemma 5, we know that if and only if . We can choose a small enough constant such that , where is given in (20). From and , we can choose small enough such that for all . Consider the following auxiliary equations: and from Lemma 1 and Lemma 2 we can get equations (24) and (25) admit globally uniformly attractive positive -periodic solution and , respectively. By the continuity of solutions with respect to parameter , for above , there exists a such that for all Define the sets as follows:

Define further the Poincaré map as follows: where is the solution of model satisfying initial condition . It follows from Theorem 1 that all solutions of model are ultimately bounded. Therefore, map is point dissipative and compact on .

We define where and for . Now, we claim In fact, from model , we directly obtain that the solution of model with initial value has the form . Hence, for all . This shows . Therefore, .

Next, we prove . Suppose that there is a such that .

We firstly have . Let be the solution of model with initial value at . From Theorem 1, we firstly have that is nonnegative for all . Furthermore, by and , from the first and fifth equations of model , we also have that and for all , respectively.

If , then by the second equation of model , we obtain

Integrating this inequality from to , we have

Further, from the third equation of model we can obtain

From the sixth equation of model , we get

Hence,

Lastly, from the equation of for model , we also have for all . Therefore, we finally obtain for all .

Similarly, if or or , we also can obtain for all . This shows that , which leads to a contradiction. This implies that . Hence, .

In , model degenerates to

We easily prove that system (36) has a globally asymptotically stable -periodic solution , where is given in Lemma 3. It is clear that map has a unique globally attractive fixed point restricted in , which is .

Now, we define , which is said to be stable set of . We show that

According to the continuity of solutions with respect to the initial values, for above given constant , there exists such that for any with , it follows that for all where is the solution of model with initial condition and . Since is -periodic, we have for any with , where is an integer. We claim that

Suppose that (38) does not hold. Then, we have

Without loss of generality, we assume that for all . Thus,

For any , let , where and are the greatest integers less than or equal to . Then, we have

It follows that , , , and for all . Thus, from model , we have the following: for any . Using the comparison principle, we obtain for any , where and are the solutions of equations (20) and (23) with respect to parameter satisfying initial conditions and , respectively.

From Lemma 1 and Lemma 2, equations (24) and (25) with respect to the parameter have globally uniformly attractive solutions and , respectively. Hence, there exists such that for all . By (26), (42), and (43), we obtain and for all . Thus, for all , it holds that